Finite Group Databasebeta

← #28349175

Subgroups — #28349175

Order 13696.

372 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×D428C211yesmaximal, characteristic, complemented
2#20292986C212yesmaximal, complemented
3#20292977C211yesmaximal, characteristic, complemented
4#20292922C212yesmaximal, complemented
5#20292907C211yesmaximal, characteristic, complemented
6#20292839C212yesmaximal
7#20292829C211yesmaximal, characteristic, complemented
8#20292758C211yesmaximal, characteristic, complemented
9#20292725C212yesmaximal, complemented
10C4×#13133803C211yesmaximal, characteristic
11#556×C107C211yesmaximal, characteristic, complemented
12C2×Q8×D107—21no—
13C2×#13133805C2212yes—
14C2×#13133803C2211yescharacteristic
15#16812109C2211yescharacteristic
16#16812109C2212yes—
17#16812089—21no—
18#16812089—22no—
19C4×D428C2211yescharacteristic
20C4×D428C2212yes—
21#16812042C2212yes—
22#16812042C2211yescharacteristic
23C4⋊(C107⋊C8)C2211yescharacteristic
24C4⋊(C107⋊C8)C2212yes—
25#16812033C2212yes—
26#16812033C2212yes—
27#16812033C2212yes—
28#16812030C2212yes—
29#16812030C2211yescharacteristic
30#16812029C2211yescharacteristic
31#16812020C2212yes—
32C4×C107⋊C8C2211yescharacteristic
33C4⋊Q8×C107C2211yescharacteristic
34#126×C107—22no—
35#124×C107C2212yes—
36C22⋊Q8×C107—21no—
37C4×Q8×C107C2212yes—
38C4×Q8×C107C2211yescharacteristic
39C4×D4×C107C2211yescharacteristic
40#117×C107—21no—
41C2×Dic214—22no—
42C2×Dic214—21no—
43Q8×D107—44no—
44C2×D428C2311yescharacteristic
45C2×C4×D107—22no—
46C2×C4×D107—21no—
47#13133805—24no—
48#13133803—22no—
49C2×C107⋊C8C2311yescharacteristic
50C2×C107⋊C8C2311yescharacteristic
51C2×C107⋊C8C2312yes—
52C4⋊Dic107—22no—
53C4⋊Dic107—21no—
54C4⋊Dic107C2312yes—
55C4⋊Dic107C2311yescharacteristic
56(C2×D107)⋊C4—22no—
57(C2×D107)⋊C4—21no—
58C2×Q8×C107D411yescharacteristic
59C2×Q8×C107C2312yes—
60C2×D4×C107C2311yescharacteristic
61C22×C4×C107—21no—
62C4⋊C4×C107—21no—
63C4⋊C4×C107—22no—
64C4⋊C4×C107—21no—
65C4⋊C4×C107D411yescharacteristic
66C4⋊C4×C107D412yes—
67C4⋊C4×C107C2311yescharacteristic
68C22⋊C4×C107—22no—
69C22⋊C4×C107—21no—
70C22⋊C4×C107—22no—
71C42×C107—22no—
72C42×C107C2311yescharacteristic
73C42×C107C2312yes—
74C23×C107—21no—
75C22×D107—21no—
76Q8×C107—22no—
77Q8×C107—22no—
78Q8×C107—22no—
79D4×C107—21no—
80D4×C107—21no—
81C2×C4×C107—21no—
82C2×C4×C107—22no—
83C2×C4×C107—42no—
84C2×C4×C107C2×D412yes—
85C2×C4×C107C2×D411yescharacteristic
86C2×C4×C107C2×D412yes—
87C2×C4×C107C2×D411yescharacteristic
88C2×C4×C107C2411yescharacteristic
89C2×Dic107—21no—
90C2×Dic107—22no—
91D428—21no—
92D428#4112yes—
93C4×D107—44no—
94C4×D107—42no—
95Dic214—44no—
96Dic214—42no—
97C107⋊C8—21no—
98C107⋊C8—22no—
99C107⋊C8—21no—
100C22×C107—41no—
101C22×C107—41no—
102C22×C107—41no—
103C22×C107C22×D411yescharacteristic
104C2×D107—22no—
105C2×D107—41no—
106C2×D107—41no—
107C4×C107—22no—
108C4×C107—22no—
109C4×C107—21no—
110C4×C107—42no—
111C4×C107—41no—
112C4×C107AGL(1,ℤ/8)12yes—
113C4×C107C2×#4111yescharacteristic
114C4×C1072−1+411yescharacteristic
115Dic107—41no—
116Dic107—42no—
117C2×C107—81no—
118C2×C107#54811yescharacteristic
119C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
120C2×C107#57811yescharacteristic
121D107—42no—
122#3367—1071nomaximal, Sylow
123C107#336711yescharacteristic, complemented, Sylow
124#556D10711yescharacteristic, complemented
125D4×Q8—1071no—
126#489—1072no—
127#487—1071no—
128#477—1072no—
129#474—1071no—
130#463—1072no—
131#461—1071no—
132#446—1071no—
133#440—1072no—
134C4×D8—1071no—
135C22×Q8—2141no—
136C2×SD16—1072no—
137C2×D8—1071no—
138C4⋊Q8—1071no—
139C4⋊Q8—1072no—
140C4⋊Q8C2×D10711yescharacteristic
141#126—22no—
142#124C2×D10712yes—
143C22⋊Q8—21no—
144C22⋊Q8—2141no—
145C22⋊Q8—2142no—
146C4×Q8C2×D10712yes—
147C4×Q8C2×D10711yescharacteristic
148C4×D4—1071no—
149C4×D4—1072no—
150C4×D4C2×D10711yescharacteristic
151#117—21no—
152#107—1071no—
153#107—1072no—
154C4⋊C8—1071no—
155C4⋊C8—1072no—
156#103—1072no—
157#103—1072no—
158#103—1072no—
159#102—1071no—
160#102—1072no—
161#102—1071no—
162#97—1072no—
163C4×C8—1071no—
164C2×Q8—4284no—
165C2×Q8—2142no—
166C2×Q8—2141no—
167C2×Q8C22⋊D10711yescharacteristic
168C2×Q8C22×D10712yes—
169C2×D4—1071no—
170C2×D4C22×D10711yescharacteristic
171C22×C4—21no—
172C22×C4—2142no—
173C22×C4—2141no—
174SD16—2144no—
175D8—2142no—
176C2×C8—1071no—
177C2×C8—1071no—
178C2×C8—1072no—
179C4⋊C4—21no—
180C4⋊C4—22no—
181C4⋊C4—21no—
182C4⋊C4—1071no—
183C4⋊C4—1072no—
184C4⋊C4—2142no—
185C4⋊C4—2141no—
186C4⋊C4C22⋊D10711yescharacteristic
187C4⋊C4C22⋊D10712yes—
188C4⋊C4C22×D10711yescharacteristic
189C22⋊C4—22no—
190C22⋊C4—21no—
191C22⋊C4—22no—
192C22⋊C4—2142no—
193C22⋊C4—2141no—
194C42—22no—
195C42C22×D10711yescharacteristic
196C42C22×D10712yes—
197C23—21no—
198C23—2141no—
199Q8—22no—
200Q8—22no—
201Q8—22no—
202Q8—4284no—
203Q8—4282no—
204D4—21no—
205D4—21no—
206D4—1072no—
207D4—2141no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—4284no—
212C2×C4—4282no—
213C2×C4—2142no—
214C2×C4—2141no—
215C2×C4C2×C22⋊D10712yes—
216C2×C4C2×C22⋊D10711yescharacteristic
217C2×C4C2×C22⋊D10712yes—
218C2×C4C2×C22⋊D10711yescharacteristic
219C2×C4C23×D10711yescharacteristic
220C8—2141no—
221C8—2142no—
222C8—2141no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—4281no—
227C22—4281no—
228C22—2142no—
229C22C22×C22⋊D10711yescharacteristic
230C4—22no—
231C4—22no—
232C4—21no—
233C4—41no—
234C4—42no—
235C4—4282no—
236C4—4281no—
237C4#1681213412yes—
238C4#41×D10711yescharacteristic
239C4#1681215611yescharacteristic
240C2—81no—
241C2—4282no—
242C2#2029328411yescharacteristic
243C2C2×#1681213411yescharacteristic
244C2#2029344211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#556×C107C2
socleC22×C107C22×D4
prime radicalC2×#13133803C22
hypercenter#556D107
hypocenterC107#3367
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. Q8×D428C2semidirect
2. #20292986C2semidirect
3. #20292977C2semidirect
4. #20292922C2semidirect
5. #20292907C2semidirect
7. #20292829C2semidirect
8. #20292758C2semidirect
9. #20292725C2semidirect
11. #556×C107C2semidirect
123. C107#3367semidirect
124. #556D107semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
Q8×D4282
#202929862
#202929772
#202929222
#202929072
#202928392
#202928292
#202927582
#202927252
C4×#131338032
#556×C1072
#3367107